PENELOPE 2003 - OECD Nuclear Energy Agency

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158 Chapter 5. Constructive quadric geometry z z x y 1,1,1,0,−1 1,1,0,0,−1 1,−1,0,0,−1 1,1,−1,0,0 1,1,−1,0,−1 1,1,−1,0,1 1,1,0,−1,0 1,0,0,−1,0 1,−1,0,−1,0 Figure 5.1: Non-planar reduced quadric surfaces and their indices [see eq. (5.13)]. In all cases, the perspective is the same as for the sphere.
5.2. Quadric surfaces 159 Table 5.1: Reduced quadrics. Reduced form Indices Quadric z − 1 = 0 0 0 0 1 −1 plane z 2 − 1 = 0 0 0 1 0 −1 pair of parallel planes x 2 + y 2 + z 2 − 1 = 0 1 1 1 0 −1 sphere x 2 + y 2 − 1 = 0 1 1 0 0 −1 cylinder x 2 − y 2 − 1 = 0 1 −1 0 0 −1 hyperbolic cylinder x 2 + y 2 − z 2 = 0 1 1 −1 0 0 cone x 2 + y 2 − z 2 − 1 = 0 1 1 −1 0 −1 one sheet hyperboloid x 2 + y 2 − z 2 + 1 = 0 1 1 −1 0 1 two sheet hyperboloid x 2 + y 2 − z = 0 1 1 0 −1 0 paraboloid x 2 − z = 0 1 0 0 −1 0 parabolic cylinder x 2 − y 2 − z = 0 1 −1 0 −1 0 hyperbolic paraboloid . . . and permutations of x, y and z that preserve the central symmetry with respect to the z-axis. For instance, this transforms the reduced sphere into an ellipsoid with semiaxes equal to the scaling factors. (ii) A rotation, R(ω, θ, φ), defined through the Euler angles OMEGA= ω, THETA= θ and PHI= φ. Notice that the rotation R(ω, θ, φ) transforms a plane perpendicular to the z-axis into a plane perpendicular to the direction with polar and azimuthal angles THETA and PHI, respectively. The first Euler angle, ω has no effect when the initial (scaled) quadric is symmetric about the z-axis. (iii) A translation, defined by the components of the displacement vector t (X-SHIFT= t x , Y-SHIFT= t y , Z-SHIFT= t z ). A quadric is completely specified by giving the set of indices (I 1 , I 2 , I 3 , I 4 , I 5 ), the scale factors (X-SCALE, Y-SCALE, Z-SCALE), the Euler angles (OMEGA, THETA, PHI) and the displacement vector (X-SHIFT, Y-SHIFT, Z-SHIFT). Any quadric surface can be expressed in this way. The implicit equation of the quadric is obtained as follows. We define the matrix ⎛ 1 A xx A 2 xy 1 A ⎞ 2 xz A = 1 ⎜ A 1 2 xy A yy A 2 yz ⎟ (5.15) ⎝ ⎠ 1 A 2 xz 1 A 2 yz A zz and write the generic quadric equation (5.12) in matrix form r T A r + A T r + A 0 = 0, (5.16)
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Data Bank ISBN 92-64-02145-0 PENELO
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FOREWORD The OECD/NEA Data Bank was
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TABLE OF CONTENTS Foreword ........
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4.3 Combined scattering and energy
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PREFACE Radiation transport in matt
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The present version of PENELOPE is
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Chapter 1 Monte Carlo simulation. B
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1.1. Elements of probability theory
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1.1. Elements of probability theory
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1.2. Random sampling methods 7 Tabl
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1.2. Random sampling methods 9 Nume
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1.2. Random sampling methods 11 whe
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1.2. Random sampling methods 13 can
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1.2. Random sampling methods 15 mus
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1.2. Random sampling methods 17 the
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1.3. Monte Carlo integration 19 Con
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1.4. Simulation of radiation transp
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¡ ¢ 1.4. Simulation of radiation
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1.5. Statistical averages and uncer
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1.6. Variance reduction 31 also imp
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1.6. Variance reduction 33 weight a
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Chapter 2 Photon interactions In th
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2.1. Coherent (Rayleigh) scattering
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2.1. Coherent (Rayleigh) scattering
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2.2. Photoelectric effect 41 ➤E E
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F 2.2. Photoelectric effect 43 1E+7
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2.3. Incoherent (Compton) scatterin
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C u 2.3. Incoherent (Compton) scatt
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2.4. Electron-positron pair product
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2.5. Attenuation coefficients 63 di
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2.6. Atomic relaxation 65 2.6 Atomi
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2.6. Atomic relaxation 67 two vacan
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Chapter 3 Electron and positron int
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3.1. Elastic collisions 71 nucleus
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3.1. Elastic collisions 73 1 E - 1
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ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ
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3.1. Elastic collisions 77 becomes
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B ' B ' 3.1. Elastic collisions 79
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3.2. Inelastic collisions 81 where
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3.2. Inelastic collisions 83 global
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3.2. Inelastic collisions 85 plays
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3.2. Inelastic collisions 87 optica
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3.2. Inelastic collisions 89 The DC
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3.2. Inelastic collisions 91 In the
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3.2. Inelastic collisions 93 where
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c b c b 3.2. Inelastic collisions 9
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A l A u 3.2. Inelastic collisions 9
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3.2. Inelastic collisions 99 After
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3.2. Inelastic collisions 101 Table
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3.2. Inelastic collisions 103 In re
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3.2. Inelastic collisions 105 1Ε+4
Page 119 and 120: 3.3. Bremsstrahlung emission 107 an
Page 121 and 122: 3.3. Bremsstrahlung emission 109 1
Page 123 and 124: 3.3. Bremsstrahlung emission 111 1
Page 125 and 126: 3.3. Bremsstrahlung emission 113 wh
Page 127 and 128: 3.3. Bremsstrahlung emission 115 Al
Page 129 and 130: 3.3. Bremsstrahlung emission 117 Th
Page 131 and 132: 3.4. Positron annihilation 119 (198
Page 133 and 134: 3.4. Positron annihilation 121 It i
Page 135 and 136: Chapter 4 Electron/positron transpo
Page 137 and 138: 4.1. Elastic scattering 125 Lewis (
Page 139 and 140: 4.1. Elastic scattering 127 ⎡ ⎛
Page 141 and 142: 4.1. Elastic scattering 129 simulat
Page 143 and 144: 4.1. Elastic scattering 131 The sim
Page 145 and 146: 4.2. Soft energy losses 133 soft in
Page 147 and 148: 4.2. Soft energy losses 135 deviati
Page 149 and 150: 4.2. Soft energy losses 137 scatter
Page 151 and 152: 4.3. Combined scattering and energy
Page 159 and 160: 4.4. Generation of random tracks 14
Page 165 and 166: Chapter 5 Constructive quadric geom
Page 167 and 168: 5.1. Rotations and translations 155
Page 169: 5.2. Quadric surfaces 157 Given a f
Page 173 and 174: 5.3. Constructive quadric geometry
Page 175 and 176: 5.4. Geometry definition file 163 1
Page 177 and 178: 5.4. Geometry definition file 165
Page 179 and 180: 5.5. The subroutine package pengeom
Page 181 and 182: 5.5. The subroutine package pengeom
Page 183 and 184: 5.6. Debugging and viewing the geom
Page 185 and 186: 5.7. A short tutorial 173 MODULE (
Page 187 and 188: 5.7. A short tutorial 175 Writing a
Page 189 and 190: Chapter 6 Structure and operation o
Page 191 and 192: 6.1. penelope 179 and/or numbers in
Page 193 and 194: 6.1. penelope 181 1986). The format
Page 195 and 196: 6.1. penelope 183 The connection of
Page 197 and 198: 6.1. penelope 185 that has to be lo
Page 199 and 200: G y y y y y 6.1. penelope 187 CALL
Page 201 and 202: 6.1. penelope 189 It is the respons
Page 203 and 204: 6.2. Examples of MAIN programs 191
Page 213 and 214: 6.3. Selecting the simulation param
Page 215 and 216: ! 6.3. Selecting the simulation par
Page 217 and 218: 6.5. Installation 205 radiation is
Page 219 and 220: 6.5. Installation 207 To get the ex
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Appendix A Collision kinematics To
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A.1. Two-body reactions 211 Clearly
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A.2. Inelastic collisions of charge
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m A.2. Inelastic collisions of char
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Appendix B Numerical tools B.1 Cubi
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B.1. Cubic spline interpolation 219
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B.2. Numerical quadrature 221 B.2 N
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Appendix C Electron/positron transp
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C.1. Tracking particles in vacuum.
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C.1. Tracking particles in vacuum.
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C.2. Exact tracking in homogeneous
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C.2. Exact tracking in homogeneous
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Bibliography Abramowitz M. and I.A.
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Bibliography 235 Briesmeister J.F.
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Bibliography 237 James F. (1990),
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Bibliography 239 Reimer L. and E.R.
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Bibliography 241 Yates A.C. (1968),
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